Properties

Like most means, the generalized mean is a homogeneous function of its arguments x_1,dots,x_n. That is, if b is a positive real number, then the generalized mean with exponent p of the numbers bcdot x_1,dots, bcdot x_n is equal to b times the generalized mean of the numbers x_1,dots, x_n.

Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks.

We get the inequality for means with exponents -p and -q, and we can use the same reasoning backwards, thus proving the inequalities to be equivalent, which will be used in some of the later proofs.

Geometric mean

For any q the inequality between mean with exponent q and geometric mean can be transformed in the following way:

prod_{i=1}^nx_i^{w_i} leq sqrt[q]{sum_{i=1}^nw_ix_i^q}

sqrt[q]{sum_{i=1}^nw_ix_i^q}leq prod_{i=1}^nx_i^{w_i}

(the first inequality is to be proven for positive q, and the latter otherwise)

We raise both sides to the power of q:

prod_{i=1}^nx_i^{w_icdot q} leq sum_{i=1}^nw_ix_i^q

in both cases we get the inequality between weighted arithmetic and geometric means for the sequence x_i^q, which can be proved by Jensen's inequality, making use of the fact the logarithmic function is concave:

sum_{i=1}^nw_ilog(x_i) leq log(sum_{i=1}^nw_ix_i)

log(prod_{i=1}^nx_i^{w_i}) leq log(sum_{i=1}^nw_ix_i)

By applying (strictly increasing) exp function to both sides we get the inequality:

since the inequality holds for any q, however small, and, as will be shown later, the expressions on the left and right approximate the geometric mean better as q approaches 0, the limit of the power mean for q approaching 0 is the geometric mean:

The proof for positive p and q is as follows:
Define the following function: f:{mathbb R_+}rightarrow{mathbb R_+},f(x)=x^{frac{q}{p}}. f is a power function, so it does have a second derivative: f(x)=(frac{q}{p})(frac{q}{p}-1)x^{frac{q}{p}-2}, which is strictly positive within the domain of f, since q > p, so we know f'' is convex.

Using this, and the Jensen's inequality we get:

f(sum_{i=1}^nw_ix_i^p)leqsum_{i=1}^nw_if(x_i^p)

sqrt[frac{p}{q}]{sum_{i=1}^nw_ix_i^p}leqsum_{i=1}^nw_ix_i^q

after raising both side to the power of 1/q (an increasing function, since 1/q is positive) we get the inequality which was to be proven:

sqrt[p]{sum_{i=1}^nw_ix_i^p}leqsqrt[q]{sum_{i=1}^nw_ix_i^q}

Using the previously shown equivalence we can prove the inequality for negative p and q by substituting them with, respectively, -q and -p, QED.

Minimum and maximum

Minimum and maximum are assumed to be the power means with exponents of
-infty and +infty. Thus for any q:

Generalized f-mean

which covers e.g. the geometric mean without using a limit. The power mean is obtained for fleft(xright)=x^p .

Applications

Signal processing

A power mean serves a non-linear moving average
which is shifted towards small signal values for small p
and emphasizes big signal values for big p.
Given an efficient implementation of a moving arithmetic mean
called smooth you can implement a moving power mean
according to the following Haskell code.